(δr i ) 2. V i. r i 2,

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1 Cartesan coordnates r, = 1, 2,... D for Eucldean space. Dstance by Pythagoras: (δs 2 = (δr 2. Unt vectors ê, dsplacement r = r ê Felds are functons of poston, or of r or of {r }. Scalar felds Φ( r, Vector felds V ( r Φ = V = 2 Φ = Φ = Φ r ê, V r, 2 Φ r 2, Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal V = jk ɛ jk V k r j ê, 3D only

2 Other smooth coordnatzaton q, = 1,... D q ( r r({q } are well defned (n some doman mostly 1 1, so Jacoban det( q / r j 0. Dstance between P {q } P {q + δq } s gven by (δs 2 = (δr k 2 = ( r k q δq r k q j δqj k k j = j g j δq δq j, where g j = r k r k q q j. k g j s a real symmetrc matrx called the metrc tensor. In general a nontrval functon of the poston, g j (q. To repeat: (δs 2 = g j δq δq j. j Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

3 Functons (felds A scalar feld f(p f( r can also be specfed by a functon of the q s, f(q = f( r(q. What about vector felds? V ( r has a meanng ndependent of the coordnates used to descrbe t, but components depend on the bass vectors. Should have bass vectors ẽ algned wth the drecton of q. How to defne? Consder r(q 1 + δq 1, q 2, q 3 r(q 1, q 2, q 3 ẽ 1 = lm = δq 1 0 δs k r k ê k q 1 g11 the smlarly defned ẽ 2 ẽ 3. In general not good orthonormal bases, because Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal ẽ 1 ẽ 2 = k r k r k q 1 q 2 / g 11 g 22 = g 12 / g 11 g 22, whch need not be zero.

4 Another problem: ẽ k s not normal to the surface q k = constant. What to do? Two approaches: Gve up on orthonormal bass vectors. Defne dfferental forms, such as dφ = Φ x k dxk. The k coeffcents of dx k transform as covarant vectors. Contravarant vectors may be consdered coeffcents of drectonal dervatves / q k. Ths s the favored approach for workng n curved spaces, dfferental geometry general relatvty. Restrct ourselves to orthogonal coordnate systems surface q = constant ntersects q j = constant at rght angles. Then q q j = 0. In general defne g j := q q j = q q j r k r k. k Note that g j s not the same as g j. Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

5 As the ê k are ndependent, ths mples A kb l = δ kl or AB T = 1I. In fact g l g lj = l l k q q l r k r k m r m r m q l q j = δ j, so g.. s the nverse matrx to g... = km q r k δ r m km q j If q q j = 0 for j, g j = 0 for j, g.. s dagonal, so g.. s also dagonal. And as (δs 2 > 0 for any non-zero δ r, g.. s postve defnte, so for an orthogonal coordnate system the dagonal elements are postve, g j = h 2 δ j, g j = h 2 δ j. Then the unt vectors are ẽ = h 1 wth the nverse relaton ê k = k h ẽ q r k =: ê k r k q =: k A k ẽ = B k ê k (1 A k B l ê l. (2 l Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

6 The set {ê k } the set {ẽ } are each orthonormal, ê k = A kẽ, so A s orthogonal, B = A. Can also check as A k A kj = h h j ( q / r k ( q j / r k = h h j g j = δ j. k Thus A can be wrtten two ways, k q A k = h r k = r k h 1 q. Note that A jk s a functon of poston, not a constant. In Eucldean space we say that ê x s the same vector regardless of whch pont r the vector s at 1. But then ẽ = A j (P ê j s not the same vector at dfferent j Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal ponts P. 1 That s, Eucldean space comes wth a prescrbed parallel transport, tellng how to move a vector wthout changng t.

7 Vector Felds So for a vector feld V (P = j Ṽ j (qẽ j = V ( rê = k V ( ra k ẽ k, Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro so Ṽk(q = V ( r(qa k (q. Also V k ( r = A k ( rṽ(q( r. Let s summarze some of our prevous relatons: ê k = A k ẽ, ẽ = A j (P ê j j Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal q A k = h r k = r k h 1 q. g j = h 2 δ j, g j = h 2 δ j

8 Gradent of a scalar feld f = f r k êk = ( f q l q l r k (A km ẽ m k klm = ( f ( q l q m q l r k h m ẽ m r k = f q l h mẽ m g ml klm lm or = lm f q l h mẽ m h 2 m δ ml = m f = m h 1 m h 1 m f q m ẽm, f q m ẽm. (3 Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal Both f( r f(q represent the same functon f(p on space, so we often carelessly leave out the twddle.

9 Velocty of a partcle v = k = j dr k dt êk = k ( dq dt h jδ j ẽ j = j r k dq q j q h j dt j h j dq j dt ẽj. r k ẽj Note t s h j dq j whch has the rght dmensons for an nfntesmal length, whle dq j by tself mght not. Example, sphercal coordnates. r sphercal shells Surfaces of constant θ cone, vertex at 0 φ plane contanng z wth the shells centered at the orgn. These ntersect at rght angles, so they are orthogonal coords. Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

10 Physcs 504, Sprng 2010 Electrcty Magnetsm By lookng at dstances from varyng one coordnate, comparng to (ds 2 = h 2 r(dr 2 + h 2 θ (dθ2 + h 2 φ (dφ2, we see that Thus h r = 1, h θ = r, h φ = r sn θ. v = ṙẽ r + r θẽ θ + r sn θ φẽ φ v 2 = ṙ 2 + r 2 θ2 + r 2 sn 2 θ φ 2. Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

11 Dervatves of Eucldean parallel transport: h 1 j q j ẽ = h 1 j = h 1 j = kl kl kl ê r j = 0 r k q j ( r k r k q j A l r k êl A kj A l r k êl. h 1 r l ê l q Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

12 Dfferental A general 1-form over a space coordnatzed by q s ω = A (P dq, s assocated wth a vector. But f we are usng orthogonal curvlnear coordnates, t s more natural to express the coeffcents as multplyng the normalzed 1-forms ω := h dq, wth ω = A dq = Ṽω. Then ω s assocated wth the vector V = Ṽẽ. Note that f ω = df = σ f q dq = h 1 f q ω, Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal the assocated vector s V = h 1 f q ẽ = f.

13 General 2-form: ω (2 = 1 B j ω ω j, wth B j = B j. 2 j In three dmensons, ths can be assocated wth a vector B = B ẽ wth B = 1 ɛ jk B jk, B jk = ɛ jk B. 2 If V ω (1 ω (2 = dω (1, then ( ω (2 = 1 B j ω ω j = d Ṽ h dq 2 = j = j j jk (Ṽh q j dq j dq h 1 Thus 1 2 B j = 1 2 h 1 h 1 j h 1 (Ṽh j q j ω j ω, ( (Ṽjh j q (Ṽh q j. Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

14 The assocated vector has coeffcents B k = 1 2 = j j ɛ jk 1 h h j ( q Ṽjh j 1 ɛ jk (Ṽj h h j q h j. q j Ṽh For cartesan coordnates h = 1, we recognze ths as the curl of V, so dω (1 V, whch s a coordnate-ndependent statement. Thus we have for any orthogonal curvlnear coordnates Ṽ ẽ = jk 1 ( ɛ jk h h j q h j Ṽ j ẽ k. (4 Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

15 B Fnally, let s consder a vector B = B ẽ ts assocated 2-form ω (2 = 1 ɛ jk B ω j ω k = 1 ɛ jk B h j h k dq j dq k. 2 2 The exteror dervatve s a three-form dω (2 = 1 2 = 1 2 = jkl jkl 1 h 1 h 2 h 3 ɛ B h j h k jk dq l dq j dq k q l ɛ jkl ɛ ljk B h j h k q l dq 1 dq 2 dq 3 ( h1 h 2 h 3 B ω 1 ω 2 ω 3. q h Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

16 A 3-form n three dmensons s assocated wth a scalar functon f(p by ω (3 = fdr 1 dr 2 dr 3 = 1 6 f abc = 1 6 f abcjk = 1 6 f abcjk ɛ abc dr a dr b dr c ɛ abc r a q r b q j r c q k dq dq j dq k ɛ abc ( = 1 6 f det A jk h 1 ω ω j ω k = f det A ω 1 ω 2 ω 3 = f ω 1 ω 2 ω 3, r a ( q h 1 j ɛ jk ω ω j ω k r b ( q j h 1 k r c q k Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal where det A = 1 because A s orthogonal, but also we assume the ẽ form a rght hed coordate system.

17 So we see that f ω (2 B, dω (2 f, wth 1 ( h1 h 2 h 3 f = B. h 1 h 2 h 3 q h In cartesan coordnates ths just reduces to B, but ths assocaton s coordnate-ndependent, so we see that n a general curvlnear coordnate system, B = 1 h 1 h 2 h 3 q ( h1 h 2 h 3 B. Fnally, let s examne the Laplacan on a scalar: f = 2 Φ = Φ 1 ( h1 h 2 h 3 = h 1 h 2 h 3 q h 1 ( h1 h 2 h 3 Φ = h 1 h 2 h 3 q q h 2 h h 1 Φ q Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

18 Summary v = dq j h j dt ẽj, j ( Ṽ ẽ = jk B = 2 Φ = f = ɛ jk 1 h h j 1 h 1 h 2 h 3 m h 1 m f q m ẽm, ( q h j Ṽ j ẽ k, ( h1 h 2 h 3 q 1 h 1 h 2 h 3 q h ( h1 h 2 h 3 h 2 B, Φ q For the record, even for generalzed coordnates that are not orthogonal, we can wrte where g := det g... 2 = 1 g j q gj g q j, Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

19 Cylndrcal Polar Although we have developed ths to deal wth esoterc orthogonal coordnate systems, such as those for your homework, let us here work out the famlar cylndrcal polar sphercal coordnate systems. Cylndrcal Polar: r, φ, z, (δs 2 = (δr 2 + r 2 (δφ 2 + (δz 2, so h r = h z = 1, h φ = r. Then f = f r ẽr + 1 f + f r φẽφ z ẽz, Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

20 Polar, contnued ( Ṽ ẽ ( Ṽ ẽ = 1 r = = 1 r ( Vz φ rv φ z ( rvφ ( Vr ẽ r + z V z r + 1 V z ẽ z r r φ ( 1 V z r φ V ( φ Vr ẽ r + z z V z r ( Vφ + r 1 V z r φ + 1 r V φ ẽ z ( (rvr r + V φ φ + (rv z z = V r r + 1 V φ r φ + V z z + 1 r V r ẽ φ ẽ φ Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

21 Polar, contnued further Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro 2 Φ = 1 r = 1 r [ ( r Φ r r + z ( r Φ r r + φ ] ( r Φ z ( 1 r Φ φ + 1 d 2 Φ r 2 dφ 2 + d2 Φ dz 2 Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

22 Sphercal Sphercal : r radus, θ polar angle, φ azmuth (δs 2 = (δr 2 +r 2 (δθ 2 +r 2 sn 2 θ(δφ 2, so h r = 1, h θ = r, h φ = r sn θ. f = f r ẽr + 1 f r θ ẽθ + 1 f r sn θ ( φẽφ V 1 = r 2 sn θ θ r sn θv φ φ rv θ ẽ r + 1 ( r sn θ φ V r r r sn θv φ + 1 ( r r rv θ θ V r ẽ φ x z ϕ θ L y ẽ θ Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

23 Sphercal, contnued V = B = [ 1 r sn θ θ (sn θv φ ] φ V θ ẽ r [ 1 + r sn θ φ V r 1 ] r r (rv φ ẽ θ + 1 [ r r (rv θ ] θ V r ẽ φ, 1 [ (r 2 r 2 sn θ sn θ r B r + (r sn θ θ B θ + (r φ B ] φ (r 2 Br + 1 = 1 r 2 r + 1 r sn θ φ B φ r sn θ (sn θ θ B θ Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal

24 Sphercal, contnued further Physcs 504, Sprng 2010 Electrcty Magnetsm Shapro 2 Φ = 1 ( r 2 sn θ r r2 sn θ Φ r + θ + 1 Φ = 1 r 2 r φ sn θ φ ( r 2 Φ + r 1 2 Φ + r 2 sn 2 θ φ 2. 1 r 2 sn θ sn θ Φ θ ( sn θ Φ θ θ Cartesan Vector Felds Dervatves Gradent Velocty of a partcle Dervatves of Dfferental Cylndrcal Polar Sphercal